Do not use $t$-SNE as predictor variables (in general).
As it has been pointed out in a previous answer in a best-case scenario using all the data to compute the $t$-SNE representation is a bad practice even if one holds out the response variable. In addition using $t$-SNE directly also renders the original modelling procedure useless when presented with new data (as the OP correctly comments too) because we cannot reconstruct a new embedding.
As a work-around I think it is worth exploring the option of a deep auto-encoder. This will allow use to readily map new data on our "interesting representation". Deep auto-encoders can learn highly non-linear representations of a dataset. We will train the auto-encoder on our training folds (similar to what we would do with PCA) and then project new data accordingly. amoeba's answer on "Building an autoencoder in Tensorflow to surpass PCA" is an excellent resource to get you started.
In specific cases, one might be able to build a deep neural network to replicate the output of $t$-SNE on a new sample. This option will take care of the above issues (data reuse, no generalisation to unseen data). I do not think it is a substantially different approach from using a deep auto-encoder in the first place though and conceptually I find it more convoluted. Plus the auto-encoder literature has many extensions (eg. denoising auto-encoders) that might come handy.
Regarding the exact phenomenon of 10%+ performance boost reported: Assuming that the response variable is not accidentally used, I suspect that the data might have an non-linear embedding that the RF classifier cannot easily detect. Random forests (or any other model for that matter) do not guarantee that any non-linear association will be discovered. I would suggest examining the $t$-SNE output itself. Maybe some of the patterns are not so hard to explain and a bit of clever/educated features engineering from your part can give you a decent boost.
I hope you see that, somewhat anti-climatically, using $t$-SNE output directly is not very straight-forward. $t$-SNE is a very helpful tool; nevertheless as many great tools, it is easy for us to misuse it as a hammer when we just have screws.